Finite permutation group/Cardinality/Fact/Proof

Proof

Let ${}M=\{1,\ldots ,n\}$. For ${}1$, there are ${}n$ possible images, for ${}2$, there are ${}n-1$ possible images remaining, for ${}3$, there are ${}n-2$ possible images remaining, etc. Therefore, there are altogether

${}n(n-1)(n-2)\cdots 2\cdot 1=n!\,$

possible permutations.

To fact